Mathematics for the interested outsider

Minkowski Space

Before we push ahead with the Faraday field in hand, we need to properly define the Hodge star in our four-dimensional space, and we need a pseudo-Riemannian metric to do this. Before we were just using the standard , but now that we’re lumping in time we need to choose a four-dimensional metric.

And just to screw with you, it will have a different signature. If we have vectors and — with time here measured in the same units as space by using the speed of light as a conversion factor — then we calculate the metric as:

In particular, if we stick the vector into the metric twice, like we do to calculate a squared-length when working with an inner product, we find:

This looks like the Pythagorean theorem in two or three dimensions, but when we get to the time dimension we subtract instead of adding them! Four-dimensional real space equipped with a metric of this form is called “Minkowski space”. More specifically, it’s called 4-dimensional Minkowski space, or “(3+1)-dimensional” Minkowski space — three spatial dimensions and one temporal dimension. Higher-dimensional versions with “spatial” dimensions (with plusses in the metric) and one “temporal” dimension (with minuses) are also called Minkowski space. And, perversely enough, some physicists write it all backwards with one plus and minuses; this version is useful if you think of displacements in time as more fundamental — and thus more useful to call “positive” — than displacements in space.

What implications does this have on the coordinate expression of the Hodge star? It’s pretty much the same, except for the determinant part. You can think about it yourself, but the upshot is that we pick up an extra factor of when the basic form going into the star involves .

So the rule is that for a basic form , the dual form consists of those component -forms not involved in , ordered such that , with a negative sign if and only if is involved in . Let’s write it all out for easy reference:

Note that the square of the Hodge star has the opposite sign from the Riemannian case; when is odd the double Hodge dual of a -form is the original form back again, but when is even the double dual is the negative of the original form.

[…] First off, I’m going to use a coordinate system where the speed of light is 1. That is, if my unit of time is seconds, my unit of distance is light-seconds. Mostly this helps keep annoying constants out of the way of the equations; physicists do this basically all the time. The other thing is that I’m going to work in four-dimensional spacetime, meaning we’ve got four coordinates: , , , and . We calculate dot products by writing . Yes, that minus sign is weird, but that’s just how spacetime works. […]

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.